MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankvalb Unicode version

Theorem rankvalb 7514
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7533 does not use Regularity, and so requires the assumption that  A is in the range of  R1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Assertion
Ref Expression
rankvalb  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Distinct variable group:    x, A

Proof of Theorem rankvalb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2830 . 2  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  _V )
2 rankwflemb 7510 . . . 4  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
3 intexrab 4207 . . . 4  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
42, 3bitri 240 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
54biimpi 186 . 2  |-  ( A  e.  U. ( R1
" On )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )
6 eleq1 2376 . . . . 5  |-  ( y  =  A  ->  (
y  e.  ( R1
`  suc  x )  <->  A  e.  ( R1 `  suc  x ) ) )
76rabbidv 2814 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
87inteqd 3904 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  y  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
9 df-rank 7482 . . 3  |-  rank  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  y  e.  ( R1 `  suc  x ) } )
108, 9fvmptg 5638 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) }  e.  _V )  ->  ( rank `  A
)  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
111, 5, 10syl2anc 642 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   E.wrex 2578   {crab 2581   _Vcvv 2822   U.cuni 3864   |^|cint 3899   Oncon0 4429   suc csuc 4431   "cima 4729   ` cfv 5292   R1cr1 7479   rankcrnk 7480
This theorem is referenced by:  rankr1ai  7515  rankidb  7517  rankval  7533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430  df-rdg 6465  df-r1 7481  df-rank 7482
  Copyright terms: Public domain W3C validator